Mechanism of cyclically polarity reversing solar magnetic cycle as a cosmic dynamo

We briefly describe historical development of the concept of solar dynamo mechanism that generates electric current and magnetic field by plasma flows inside the solar convection zone. The dynamo is the driver of the cyclically polarity reversing solar magnetic cycle. The reversal process can easily and visually be understood in terms of magnetic field line stretching and twisting and folding in three-dimensional space by plasma flows of differential rotation and global convection under influence of Coriolis force. This process gives rise to formation of a series of huge magnetic flux tubes that propagate along iso-rotation surfaces inside the convection zone. Each of these flux tubes produces one solar cycle. We discuss general characteristics of any plasma flows that can generate magnetic field and reverse the polarity of the magnetic field in a rotating body in the Universe. We also mention a list of problems which are currently being disputed concerning the solar dynamo mechanism together with observational evidences that are to be constraints as well as verifications of any solar cycle dynamo theories of short and long term behaviors of the Sun, particularly time variations of its magnetic field, plasma flows, and luminosity.     

The Sun’s Large-Scale Magnetic Field and Its Long-Term Evolution

The Sun’s large-scale external field is formed through the emergence of magnetic flux in active regions and its subsequent dispersal over the solar surface by differential rotation, supergranular convection, and meridional flow. The observed evolution of the polar fields and open flux (or interplanetary field) during recent solar cycles can be reproduced by assuming a supergranular diffusion rate of 500 – 600 km² s⁻¹ and a poleward flow speed of 10 –20 m s⁻¹. The nonaxisymmetric component of the large-scale field decays on the flow timescale of ∼1 yr and must be continually regenerated by new sunspot activity. Stochastic fluctuations in the longitudinal distribution of active regions can produce large peaks in the Sun’s equatorial dipole moment and in the interplanetary field strength during the declining phase of the cycle; by the same token, they can lead to sudden weakenings of the large-scale field near sunspot maximum (Gnevyshev gaps). Flux transport simulations over many solar cycles suggest that the meridional flow speed is correlated with cycle amplitude, with the flow being slower during less active cycles.     

Global solar dynamo models: Simulations and predictions

Flux-transport type solar dynamos have achieved considerable success in correctly simulating many solar cycle features, and are now being used for prediction of solar cycle timing and amplitude. We first define flux-transport dynamos and demonstrate how they work. The essential added ingredient in this class of models is meridional circulation, which governs the dynamo period and also plays a crucial role in determining the Sun’s memory about its past magnetic fields. We show that flux-transport dynamo models can explain many key features of solar cycles. Then we show that a predictive tool can be built from this class of dynamo that can be used to predict mean solar cycle features by assimilating magnetic field data from previous cycles.     

A comprehensive view of solar-terrestrial relationships in terms of a chain of four dynamo-powered plasma acceleration processes

This paper emphasizes an effort to link processes which relate solar activity and magnetospheric disturbances in terms of energy transfer through a chain of four elements. In this view, each element is explicitly thought to be powered by a dynamo, namely the solar wind generation dynamo, the solar flare dynamo, the solar wind-magnetosphere dynamo and the auroral dynamo, respectively. Each dynamo powers a plasma acceleration process by the Lorentz force and the plasma flows thus generated are the solar wind, the flare-generated solar wind disturbance, the magnetospheric plasma convection and the ionospheric convection, respectively. Each plasma flow conveys the energy from one element to the next in the chain. Some of the kinetic energy of the photospheric plasma is eventually deposited in the polar ionosphere as heat energy.     

Differential rotation and meridional flow of Arcturus

The spectroscopic variability of Arcturus hints at cyclic activity cycle and differential rotation. This could provide a test of current theoretical models of solar and stellar dynamos. To examine the applicability of current models of the flux transport dynamo to Arcturus, we compute a mean‐field model for its internal rotation, meridional flow, and convective heat transport in the convective envelope. We then compare the conditions for dynamo action with those on the Sun. We find solar‐type surface rotation with about 1/10th of the shear found on the solar surface. The rotation rate increases monotonically with depth at all latitudes throughout the whole convection zone. In the lower part of the convection zone the horizontal shear vanishes and there is a strong radial gradient. The surface meridional flow has maximum speed of 170 m/s and is directed towards the equator at high and towards the poles at low latitudes. Turbulent magnetic diffusivity is of the order 10¹⁵–10¹⁶ cm²/s. The conditions on Arcturus are not favorable for a circulation‐dominated dynamo (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A joined model for solar dynamo and differential rotation

A model for the solar dynamo, consistent in global flow and numerical method employed with the differential rotation model, is developed. The magnetic turbulent diffusivity is expressed in terms of the entropy gradient, which is controlled by the model equations. The magnetic Prandtl number and latitudinal profile of the alpha-effect are specified by fitting the computed period of the activity cycle and the equatorial symmetry of magnetic fields to observations. Then, the instants of polar field reversals and time-latitude diagrams of the fields also come into agreement with observations. The poloidal field has a maximum amplitude of about 10 Gs in the polar regions. The toroidal field of several thousand Gauss concentrates near the base of the convection zone and is transported towards the equator by the meridional flow. The model predicts a value of about 10³⁷ erg for the total magnetic energy of large-scale fields in the solar convection zone.     

Non‐holonomic filamentary dynamos as generalized Arnold's maps in Riemannian space

A filamentary non‐holonomic dynamo solution of self‐induction magnetic field equation is found by considering highly conducting filaments. It is shown that planar filaments cannot support dynamo action since the flow along the filament vanishes for torsion‐free filaments. This is a generalization of the Zeldovich theorem for linear magnetic dynamo filaments. The flow of filament is proportionally to the product between Frenet torsion and curvature. This shows that filamentary dynamos must possess Frenet torsion. A well‐known example of this result is the α ‐dynamo in solar physics. Magnetic helicity and magnetic energy for this filamentary dynamo are computed. Magnetic helicity vanishes by construction and the magnetic field decays with torsion energy in helicoidal dynamos. The approach considered here is useful for the investigation of anisotropic turbulent cascades. As a particular simple example it is shown that under certain constraints the solution can be reduced to the Arnold cat dynamo map solution where the non‐holonomic directional mixed derivative, would play the role of the Lyapunov exponent which appears on stretching the magnetic field in Riemannian space. The solution seems to describe marginal slow dynamos when the velocities involved in the dynamo flows are constants. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of the effect of a mean velocity field on a mean field dynamo

We study semi-analytically and in a consistent manner the generation of a mean velocity field     by helical magnetohydrodynamical (MHD) turbulence, and the effect that this field can have on a mean field dynamo. Assuming a prescribed, maximally helical small-scale velocity field, we show that large-scale flows can be generated in MHD turbulent flows via small-scale Lorentz force. These flows back-react on the mean electromotive force of a mean field dynamo through new terms, leaving the original α and β terms explicitly unmodified. Cross-helicity plays the key role in interconnecting all the effects. In the minimal τ closure that we chose to work with, the effects are stronger for large relaxation times.     

The Kinematic Theory of Solar Dynamo

Generation of the Sun‘s magnetic fields by self-inductive processes in the solar electrically conducting interior, the solar dynamo theory, is a fundamentally important subject in astrophysics. The kinematic dynamo theory concerns how the magnetic fields are produced by kinematically possible flows without being constrained by the dynamic equation. We review a number of basic aspects of the kinematic dynamo theory, including the magnetohydrodynamic approximation for the dynamo equation, the impossibility of dynamo action with the solar differential rotation, the Cowling‘s anti-dynamo theorem in the solar context, the turbulent alpha effect and recently constructed three-dimensional interface dynamos controlled by the solar tachocline at the base of the convection zone.     

Torsional Oscillations in a Global Solar Dynamo

We characterize and analyze rotational torsional oscillations developing in a large-eddy magnetohydrodynamical simulation of solar convection (Ghizaru, Charbonneau, and Smolarkiewicz, Astrophys. J. Lett. 715, L133, 2010; Racine et al., Astrophys. J. 735, 46, 2011) producing an axisymmetric, large-scale, magnetic field undergoing periodic polarity reversals. Motivated by the many solar-like features exhibited by these oscillations, we carry out an analysis of the large-scale zonal dynamics. We demonstrate that simulated torsional oscillations are not driven primarily by the periodically varying large-scale magnetic torque, as one might have expected, but rather via the magnetic modulation of angular-momentum transport by the large-scale meridional flow. This result is confirmed by a straightforward energy analysis. We also detect a fairly sharp transition in rotational dynamics taking place as one moves from the base of the convecting layers to the base of the thin tachocline-like shear layer formed in the stably stratified fluid layers immediately below. We conclude by discussing the implications of our analyses with regard to the mechanism of amplitude saturation in the global dynamo operating in the simulation, and speculate on the possible precursor value of torsional oscillations for the forecast of solar-cycle characteristics.     

On the interaction between differential rotation and magnetic fields in the Sun

We have performed 3-D numerical simulations of compressible convection under the influence of rotation and magnetic fields in spherical shells. They aim at understanding the subtle coupling between convection, rotation and magnetic fields in the solar convection zone. We show that as the magnetic Reynolds number is increased in the simulations, the magnetic energy saturates via nonlinear dynamo action, to a value smaller but comparable to the kinetic energy contained in the shell, leading to increasingly strong Maxwell stresses that tend to weaken the differential rotation driven by the convection. These simulations also indicate that the mean toroidal and poloidal magnetic fields are small compared to their fluctuating counterparts, most of the magnetic energy being contained in the non-axisymmetric fields. The intermittent nature of the magnetic fields generated by such a turbulent convective dynamo confirms that in the Sun the large-scale ordered dynamo responsible for the 22-year cycle of activity can hardly be located in the solar convective envelope.     

Large-scale flows excited by magnetic fields in the solar convective zone

Observations demonstrate a nearly 22-year periodic zonal flow superimposed on general solar differential rotation (LaBonte and Howard, 1982) and some meridional motions (e.g., Tuominen, Tuominen, and Kyrolänen, 1983). Such flows can be excited by the magnetic wave generated by the dynamo in the solar convective zone.An approximate analytical solution for the zonal and meridional flows for a given magnetic wave is constructed. This approach is justified by the fact that the magnetic field is generated by differential rotation and mean helicity, and the magnetic field in the time interval under consideration does not affect much this main flow; it can, however, strongly influence the perturbations of this flow.The density gradient in the convective zone is taken into account as an essential point in the solution construction. The solution agreed well with observational features and, in particular, it gives a phase shift between the rotational (zonal) wave and solar activity. A polar branch of the rotational wave can be described as an effect created by a poleward moving dynamo wave.Secular variations in the symmetrical part of the differential rotation and in the asymmetry between the north and south hemispheres are predicted.The alternative approaches to the explanation of the origin of the observed large-scale flows are discussed.     

Quenching of Meridional Circulation in Flux Transport Dynamo Models

Guided by the recent observational result that the meridional circulation of the Sun becomes weaker at the time of the sunspot maximum, we have included a parametric quenching of the meridional circulation in solar dynamo models such that the meridional circulation becomes weaker when the magnetic field at the base of the convection zone is stronger. We find that a flux transport solar dynamo tends to become unstable on including this quenching of meridional circulation if the diffusivity in the convection zone is less than about 2×10¹¹ cm² s⁻¹. The quenching of α, however, has a stabilizing effect and it is possible to stabilize a dynamo with low diffusivity with sufficiently strong α-quenching. For dynamo models with high diffusivity, the quenching of meridional circulation does not produce a large effect and the dynamo remains stable. We present a solar-like solution from a dynamo model with diffusivity 2.8×10¹² cm² s⁻¹ in which the quenching of meridional circulation makes the meridional circulation vary periodically with solar cycle as observed and does not have any other significant effect on the dynamo.     

On the Connection Between Mean Field Dynamo Theory and Flux Tubes

Mean field dynamo theory deals with various mean quantities and does not directly throw any light on the question of existence of flux tubes. We can, however, draw important conclusions about flux tubes in the interior of the Sun by combining additional arguments with the insights gained from solar dynamo solutions. The polar magnetic field of the Sun is of order 10 G, whereas the toroidal magnetic field at the bottom of the convection zone has been estimated to be 100000 G. Simple order-of-magnitude estimates show that the shear in the tachocline is not sufficient to stretch a 10 G mean radial field into a 100000 G mean toroidal field. We argue that the polar field of the Sun must get concentrated into intermittent flux tubes before it is advected to the tachocline. We estimate the strengths and filling factors of these flux tubes. Stretching by shear in the tachocline is then expected to produce a highly intermittent magnetic configuration at the bottom of the convection zone. The meridional flow at the bottom of the convection zone should be able to carry this intermittent magnetic field equatorward, as suggested recently by Nandy and Choudhuri (2002). When a flux tube from the bottom of the convection zone rises to a region of pre-existing poloidal field at the surface, we point out that it picks up a twist in accordance with the observations of current helicities at the solar surface.     

An essay on sunspots and solar flares

The presently prevailing theories of sunspots and solar flares rely on the hypothetical presence of magnetic flux tubes beneath the photosphere and the two subsequent hypotheses, their emergence above the photosphere and explosive magnetic reconnection, converting magnetic energy carried by the flux tubes for solar flare energy.In this paper, we pay attention to the fact that there are large-scale magnetic fields which divide the photosphere into positive and negative (line-of-sight) polarity regions and that they are likely to be more fundamental than sunspot fields, as emphasized most recently by McIntosh (1981). A new phenomenological model of the sunspot pair formation is then constructed by considering an amplification process of these largescale fields near their boundaries by shear flows, including localized vortex motions. The amplification results from a dynamo process associated with such vortex flows and the associated convergence flow in the largescale fields.This dynamo process generates also some of the familiar “force-free” fields or the “sheared” magnetic fields in which the magnetic field-aligned currents are essential. Upward field-aligned currents generated by the dynamo process are carried by downward streaming electrons which are expected to be accelerated by an electric potential structure; a similar structure is responsible for accelerating auroral electrons in the magnetosphere. Depending on the magnetic field configuration and the shear flows, the current-carrying electrons precipitate into different geometrical patterns, causing circular flares, umbral flares, two-ribbon flares, etc. Thus, it is suggested that “low temperature flares” are directly driven by the photospheric dynamo process.     

The role of the Yoshizawa effect in the Archontis dynamo

The generation of mean magnetic fields is studied for a simple non-helical flow where a net cross-helicity of either sign can emerge. This flow, which is also known as the Archontis flow, is a generalization of the Arnold–Beltrami–Childress flow, but with the cosine terms omitted. The presence of cross-helicity leads to a mean-field dynamo effect that is known as the Yoshizawa effect. Direct numerical simulations of such flows demonstrate the presence of magnetic fields on scales larger than the scale of the flow. Contrary to earlier expectations, the Yoshizawa effect is found to be proportional to the mean magnetic field and can therefore lead to its exponential instead of just linear amplification for magnetic Reynolds numbers that exceed a certain critical value. Unlike α effect dynamos, it is found that the Yoshizawa effect is not notably constrained by the presence of a conservation law. It is argued that this is due to the presence of a forcing term in the momentum equation, which leads to a non-zero correlation with the magnetic field. Finally, the application to energy convergence in solar wind turbulence is discussed.     

Formation of polar starspots through meridional circulation

To explain the observed intermingling of polarities in the magnetic field distributions of rapidly rotating stars, surface magnetic flux transport models demand the presence of fast meridional flows.We combine simulations of the pre-eruptive and post-eruptive magnetic flux transport in cool stars to investigate the influence of a fast meridional circulation on the latitudinal eruption pattern of magnetic flux tubes and on the polar magnetic field properties. Magnetic flux tubes rising through the convection zone experience an enhanced latitude-dependent poleward deflection through meridional flows, which renders the wings of stellar butterfly diagrams convex. The larger amount of magnetic flux emerging at higher latitudes supports the intermingling of opposite polarities of polar magnetic fields and yields magnetic flux densities in the polar regions about 20% higher than in the case disregarding the pre-eruptive deflection. Taking the pre-eruptive evolution of magnetic flux into account therefore eases the need for the fast meridional flows predicted by previous investigations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential rotation and meridional circulation in global models of solar convection

In the outer envelope of the Sun and in other stars, differential rotation and meridional circulation are maintained via the redistribution of momentum and energy by convective motions. In order to properly capture such processes in a numerical model, the correct spherical geometry is essential. In this paper I review recent insights into the maintenance of mean flows in the solar interior obtained from high-resolution simulations of solar convection in rotating spherical shells. The Coriolis force induces a Reynolds stress which transports angular momentum equatorward and also yields latitudinal variations in the convective heat flux. Meridional circulations induced by baroclinicity and rotational shear further redistribute angular momentum and alter the mean stratification. This gives rise to a complex nonlinear interplay between turbulent convection, differential rotation, meridional circulation, and the mean specific entropy profile. I will describe how this drama plays out in our simulations as well as in solar and stellar convection zones. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simple laminar dynamos

Although current interest in astrophysical dynamo theory is largely focussed on flows with both large‐ and small‐scale motions, historically the study of dynamos driven by laminar flows has been important. Some classical laminar flow dynamos are reviewed. These results were obtained in an asymptotic regime corresponding to small values of system parameters. Numerical simulations have since been used to extend these results outside of these asymptotic regimes; the asymptotic results remain useful approximations well outside of their formal regions of validity. By changing slightly the system geometries some interesting new results have recently been obtained The latter include the very simple “one‐roll” dynamo, with motions in a single meridional cell contained within a spherical volume of fluid, without differential rotation (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The impact of meridional circulation on stellar butterfly diagrams and polar caps

Observations of rapidly rotating solar-like stars show a significant mixture of opposite-polarity magnetic fields within their polar regions. To explain these observations, models describing the surface transport of magnetic flux demand the presence of fast meridional flows. Here, we link subsurface and surface magnetic flux transport simulations to investigate (i) the impact of meridional circulations with peak velocities of  ≤125 m s⁻¹  on the latitudinal eruption pattern of magnetic flux tubes and (ii) the influence of the resulting butterfly diagrams on polar magnetic field properties. Prior to their eruption, magnetic flux tubes with low field strengths and initial cross-sections below  ∼300 km  experience an enhanced poleward deflection through meridional flows (assumed to be polewards at the top of the convection zone and equatorwards at the bottom). In particular, flux tubes which originate between low and intermediate latitudes within the convective overshoot region are strongly affected. This latitude-dependent poleward deflection of erupting magnetic flux renders the wings of stellar butterfly diagrams distinctively convex. The subsequent evolution of the surface magnetic field shows that the increased number of newly emerging bipoles at higher latitudes promotes the intermingling of opposite polarities of polar magnetic fields. The associated magnetic flux densities are about 20 per cent higher than in the case disregarding the pre-eruptive deflection, which eases the necessity for fast meridional flows predicted by previous investigations. In order to reproduce the observed polar field properties, the rate of the meridional circulation has to be of the order of 100 m s⁻¹, and the latitudinal range from which magnetic flux tubes originate at the base of the convective zone (≲50°) must be larger than in the solar case (≲35°).